Average Error: 29.4 → 22.3
Time: 11.9s
Precision: 64
\[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
↓
\[\begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.62963224260069623 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 8.0651678035819246 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]
{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}
↓
\begin{array}{l}
\mathbf{if}\;\frac{1}{n} \le -2.62963224260069623 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 8.0651678035819246 \cdot 10^{-8}\right):\\
\;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\

\mathbf{else}:\\
\;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\

\end{array}
double f(double x, double n) {
        double r78731 = x;
        double r78732 = 1.0;
        double r78733 = r78731 + r78732;
        double r78734 = n;
        double r78735 = r78732 / r78734;
        double r78736 = pow(r78733, r78735);
        double r78737 = pow(r78731, r78735);
        double r78738 = r78736 - r78737;
        return r78738;
}
↓
double f(double x, double n) {
        double r78739 = 1.0;
        double r78740 = n;
        double r78741 = r78739 / r78740;
        double r78742 = -2.629632242600696e-07;
        bool r78743 = r78741 <= r78742;
        double r78744 = 8.065167803581925e-08;
        bool r78745 = r78741 <= r78744;
        double r78746 = !r78745;
        bool r78747 = r78743 || r78746;
        double r78748 = x;
        double r78749 = r78748 + r78739;
        double r78750 = pow(r78749, r78741);
        double r78751 = pow(r78748, r78741);
        double r78752 = r78750 - r78751;
        double r78753 = exp(r78752);
        double r78754 = log(r78753);
        double r78755 = r78741 / r78748;
        double r78756 = 0.5;
        double r78757 = r78756 / r78740;
        double r78758 = 2.0;
        double r78759 = pow(r78748, r78758);
        double r78760 = r78757 / r78759;
        double r78761 = log(r78748);
        double r78762 = r78761 * r78739;
        double r78763 = pow(r78740, r78758);
        double r78764 = r78748 * r78763;
        double r78765 = r78762 / r78764;
        double r78766 = r78760 - r78765;
        double r78767 = r78755 - r78766;
        double r78768 = r78747 ? r78754 : r78767;
        return r78768;
}

Error

Bits error versus x

Bits error versus n

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 2 regimes

  2. if (/ 1.0 n) < -2.629632242600696e-07 or 8.065167803581925e-08 < (/ 1.0 n)

    1. Initial program 9.0

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]
    2. Using strategy rm

    3. Applied add-log-exp9.2

      \[\leadsto {\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - \color{blue}{\log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)}\]

    4. Applied add-log-exp9.2

      \[\leadsto \color{blue}{\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}\right)} - \log \left(e^{{x}^{\left(\frac{1}{n}\right)}}\right)\]

    5. Applied diff-log9.2

      \[\leadsto \color{blue}{\log \left(\frac{e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)}}}{e^{{x}^{\left(\frac{1}{n}\right)}}}\right)}\]

    6. Simplified9.2

      \[\leadsto \log \color{blue}{\left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)}\]

    if -2.629632242600696e-07 < (/ 1.0 n) < 8.065167803581925e-08

    1. Initial program 44.4

      \[{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}\]

    2. Taylor expanded around inf 32.5

      \[\leadsto \color{blue}{1 \cdot \frac{1}{x \cdot n} - \left(0.5 \cdot \frac{1}{{x}^{2} \cdot n} + 1 \cdot \frac{\log \left(\frac{1}{x}\right)}{x \cdot {n}^{2}}\right)}\]

    3. Simplified32.0

      \[\leadsto \color{blue}{\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification22.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{1}{n} \le -2.62963224260069623 \cdot 10^{-7} \lor \neg \left(\frac{1}{n} \le 8.0651678035819246 \cdot 10^{-8}\right):\\ \;\;\;\;\log \left(e^{{\left(x + 1\right)}^{\left(\frac{1}{n}\right)} - {x}^{\left(\frac{1}{n}\right)}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{1}{n}}{x} - \left(\frac{\frac{0.5}{n}}{{x}^{2}} - \frac{\log x \cdot 1}{x \cdot {n}^{2}}\right)\\ \end{array}\]

Reproduce

herbie shell --seed 2020020 
(FPCore (x n)
  :name "2nthrt (problem 3.4.6)"
  :precision binary64
  (- (pow (+ x 1) (/ 1 n)) (pow x (/ 1 n))))